The Average Size of 2-Selmer Groups of Elliptic Curves over Function Fields

Abstract: Given an elliptic curve $E$ over a global field $K$, the abelian group $E(K)$ is finitely generated, and so much effort has been put into trying to understand the behavior of $\operatorname{rank}E(K)$, as $E$ varies. Of note, it is a folklore conjecture that, when all elliptic curves $E/K$ are ordered by a suitably defined height, the average value of their ranks is exactly $1/2$. One fruitful avenue for understanding the distribution of $\operatorname{rank}E(K)$ has been to first understand the distribution of the sizes of Selmer groups of elliptic curves. In this direction, various authors (including Bhargava-Shankar, Poonen-Rains, and Bhargava-Kane-Lenstra-Poonen-Rains) have made conjectures which predict, for example, that the average size of the $n$-Selmer group of $E/K$ is equal to the sum of the divisors of $n$. In this talk, I will report on some recent work verifying this average size prediction, "up to small error term," whenever $n=2$ and $K$ is any global *function* field. Results along these lines were previously known whenever $K$ was a number field or function field of characteristic $\ge 5$, so the novelty of my work is that it applies even in "bad" characteristic.

number theory

Audience: researchers in the topic

Harvard number theory seminar